Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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56
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9answers
8k views

Is infinity a number?

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school ...
24
votes
3answers
4k views

difference between class, set , family and collection

In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
278
votes
18answers
51k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
21
votes
3answers
1k views

Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) ...
17
votes
3answers
1k views

Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?

I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...
29
votes
13answers
3k views

What exactly is infinity?

On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
3
votes
2answers
219 views

Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”? [duplicate]

Possible Duplicate: Alternative ways to say “if and only if”? So when I come across mathematical definitions like "A function is continuous if...."A space is compact ...
51
votes
9answers
3k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
42
votes
10answers
3k views

Given real numbers: define integers?

I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following: Integer numbers are just special cases (a subset) of real numbers. ...
25
votes
8answers
2k views

Why does the Dedekind Cut work well enough to define the Reals?

I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals. I just can't get the ...
11
votes
3answers
2k views

How can an ordered pair be expressed as a set?

My book says \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation} I have been staring at this for a bit and it is not making since to me. I have read several others posts on this, but none made ...
7
votes
4answers
1k views

Alternate definition of prime number

I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works: A prime is a quantity $p$ such that whenever $p$ is a factor of ...
26
votes
11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
56
votes
14answers
3k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
11
votes
8answers
1k views

Why do we accept Kuratowski's definition of ordered pairs?

I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly ...
18
votes
7answers
2k views

Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
15
votes
4answers
578 views

What's wrong with this “backwards” definition of limit?

Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?: $\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if ...
7
votes
1answer
468 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
6
votes
7answers
573 views

What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
6
votes
6answers
1k views

Definition of an Ordered Pair

"The ordered pair $(a,b)$ is defined to be the set $\{\{a\},\{a,b\}\}$." ~ Hungerford's Algebra (p.6) I think this is the first time that i've seen this definition. I've read the wiki page. Is it ...
2
votes
3answers
246 views

A question about a certain way to define mathematical objects

It is common in mathematics to see definitions of the following form: we begin with a certain object $A$. we perform some construction depending on a choice of some parameter $\lambda\in\Lambda$ for ...
6
votes
5answers
176 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
14
votes
6answers
737 views

How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity?

I'm told that a function defined on an interval $[a,b]$ or $(a,b)$ is uniformly continuous if for each $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that $|x-t|\lt \delta$ implies that ...
10
votes
2answers
424 views

True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
7
votes
7answers
1k views

Infinite limits

Does a limit that has the value of infinite exist or not? I've recently come across certain sources that say that if the value of a limit is infinite, then that limit does not exist. This contradicts ...
18
votes
9answers
1k views

Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
2
votes
3answers
195 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
10
votes
3answers
462 views

What is the purpose of the $\mp$ symbol in mathematical usage?

Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please ...
3
votes
3answers
693 views

Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
1
vote
2answers
450 views

Is this Vector operation defined? Does it have a name?

Let's say I have 2 vectors: [a, b, c] [x, y, z] And I need to do an operation like the following for a computer program: ...
17
votes
12answers
3k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
18
votes
5answers
2k views

What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
16
votes
5answers
1k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
18
votes
3answers
1k views

GCD of rationals

Disclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ ...
10
votes
1answer
4k views

Difference between a theorem and a law

There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra: Theorems Idempotent Involution Theorem of Complementarity Laws Commutative ...
1
vote
4answers
1k views

Why aren't logarithms defined for negative $x$?

Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...
9
votes
3answers
298 views

How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
10
votes
8answers
10k views

What is the difference between equation and formula?

Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference. ...
8
votes
4answers
610 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
7
votes
1answer
466 views

Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1?

I am trying to identify what the flaw is exactly when reasoning about a limit such as the definition of $\mathbf e$: $$ \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n={e} $$ Now, I know ...
6
votes
2answers
806 views

Precise definition of “weaker” and “stronger”?

If I say that $A$ is stronger than $B$, do I mean that $A \Rightarrow B$, or that $B \Rightarrow A$? (Or something else?) I feel like I have seen both usages in literature, which is confusing. ...
4
votes
3answers
968 views

Two definitions of $\limsup$

Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$: Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n = ...
8
votes
3answers
868 views

Is the empty graph connected?

Is the empty graph always connected ? I've looked through some sources (for example Diestels "Graph theory") and this special case seems to be ommited. What is the general opinion for this case ? As ...
4
votes
5answers
510 views

What is a Real Number?

I believe I'm over thinking it, but I want to be 100% sure. A Real Number is any number, correct? Whether it be an integer or something else. It's the set $\mathbb R$ from $(-\infty, +\infty)$ ...
4
votes
1answer
533 views

Definitions for limsup and liminf

I was wondering what are the general spaces that the concepts limsup and liminf can apply to? Is complete lattice one of them? Also How about metric space? What are limsup and liminf specified with ...
3
votes
1answer
114 views

Alternative definition of strong/weak operator topology.

Given two normed spaces $(X, ||\cdot||_X)$ and $(Y,||\cdot||_Y)$ the space of bounded linear maps $\mathcal{B}(X,Y)$ can be equipped with the strong operator topology (SOT) as follows: The ...
1
vote
2answers
62 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
1
vote
1answer
123 views

Field of sets versus a field as an algebraic structure

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
15
votes
2answers
531 views

What is it to be normal?

I'm interested to find out why the word 'normal' crops up so many different areas of mathematics, especially but not exclusively in abstract algebra, and how the definitions are related, if at all. ...
8
votes
3answers
348 views

Why is closure omitted in some group definitions?

In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?